Algebraic & Geometric Topology

A rational splitting of a based mapping space

Katsuhiko Kuribayashi and Toshihiro Yamaguchi

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Abstract

Let (X,Y) be the space of base-point-preserving maps from a connected finite CW complex X to a connected space Y. Consider a CW complex of the form Xαek+1 and a space Y whose connectivity exceeds the dimension of the adjunction space. Using a Quillen–Sullivan mixed type model for a based mapping space, we prove that, if the bracket length of the attaching map α:SkX is greater than the Whitehead length WL(Y) of Y, then (Xαek+1,Y) has the rational homotopy type of the product space (X,Y)×Ωk+1Y. This result yields that if the bracket lengths of all the attaching maps constructing a finite CW complex X are greater than WL(Y) and the connectivity of Y is greater than or equal to dimX, then the mapping space (X,Y) can be decomposed rationally as the product of iterated loop spaces.

Article information

Source
Algebr. Geom. Topol., Volume 6, Number 1 (2006), 309-327.

Dates
Received: 19 July 2005
Revised: 14 February 2006
Accepted: 14 February 2006
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.agt/1513796514

Digital Object Identifier
doi:10.2140/agt.2006.6.309

Mathematical Reviews number (MathSciNet)
MR2220679

Zentralblatt MATH identifier
1097.55010

Subjects
Primary: 55P62: Rational homotopy theory
Secondary: 54C35: Function spaces [See also 46Exx, 58D15]

Keywords
mapping space $d_1$–depth bracket length Whitehead length

Citation

Kuribayashi, Katsuhiko; Yamaguchi, Toshihiro. A rational splitting of a based mapping space. Algebr. Geom. Topol. 6 (2006), no. 1, 309--327. doi:10.2140/agt.2006.6.309. https://projecteuclid.org/euclid.agt/1513796514


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